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Graded multiplicity in harmonic polynomials from the Vinberg setting

Alexander Heaton

Abstract

We consider Vinberg $θ$-groups associated to a cyclic quiver on $r$ nodes. Let $K$ be the product of general linear groups associated to the nodes, acting naturally on $V = \oplus \text{Hom}(V_i, V_{i+1})$. We study the harmonic polynomials on $V$ in the specific case where $\dim V_i = 2$ for all $i$. For each multigraded component of the harmonics, we give an explicit decomposition into irreducible representations of $K$, and additionally describe the multiplicities of each irreducible by counting integral points on certain faces of a polyhedron.

Graded multiplicity in harmonic polynomials from the Vinberg setting

Abstract

We consider Vinberg -groups associated to a cyclic quiver on nodes. Let be the product of general linear groups associated to the nodes, acting naturally on . We study the harmonic polynomials on in the specific case where for all . For each multigraded component of the harmonics, we give an explicit decomposition into irreducible representations of , and additionally describe the multiplicities of each irreducible by counting integral points on certain faces of a polyhedron.

Paper Structure

This paper contains 4 sections, 14 theorems, 56 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background from some existing literature
  3. Preliminary setup and main results
  4. Proof of Theorem \ref{['thm:harmonics-character']}

Key Result

Proposition 1

Figures (1)

  • Figure 1: $\text{dim Hom}_K(F_{z,s},\mathcal{H}_n) = 6$ for $r=3$, $n=(6,5,3)$, $s = (7,5,4)$, and $z=(5,1)$.

Theorems & Definitions (28)

  • Proposition 1
  • proof
  • Theorem 1
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2
  • proof
  • Proposition 2
  • ...and 18 more